mathlib3 documentation

category_theory.functor.basic

Functors #

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Defines a functor between categories, extending a prefunctor between quivers.

Introduces notation C ⥤ D for the type of all functors from C to D. (Unfortunately the ⇒ arrow (\functor) is taken by core, but in mathlib4 we should switch to this.)

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def category_theory.functor.to_prefunctor {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (self : C ⥤ D) :
C ⥤q D

The prefunctor between the underlying quivers.

Instances for category_theory.functor.to_prefunctor
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structure category_theory.functor (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
Type (max v₁ v₂ u₁ u₂)

functor C D represents a functor between categories C and D.

To apply a functor F to an object use F.obj X, and to a morphism use F.map f.

The axiom map_id expresses preservation of identities, and map_comp expresses functoriality.

See https://stacks.math.columbia.edu/tag/001B.

Instances for category_theory.functor
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@[simp]
theorem category_theory.functor.map_id {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (self : C ⥤ D) (X : C) :
self.map (𝟙 X) = 𝟙 (self.obj X)
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@[simp]
theorem category_theory.functor.map_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (self : C ⥤ D) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :
self.map (f ≫ g) = self.map f ≫ self.map g
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theorem category_theory.functor.map_comp_assoc {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (self : C ⥤ D) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) {X' : D} (f' : self.obj Z ⟶ X') :
self.map (f ≫ g) ≫ f' = self.map f ≫ self.map g ≫ f'
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@[protected]
def category_theory.functor.id (C : Type u₁) [category_theory.category C] :
C ⥤ C

𝟭 C is the identity functor on a category C.

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Instances for category_theory.functor.id
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@[protected, instance]
def category_theory.functor.inhabited (C : Type u₁) [category_theory.category C] :
inhabited (C ⥤ C)
Equations
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@[simp]
theorem category_theory.functor.id_obj {C : Type u₁} [category_theory.category C] (X : C) :
(𝟭 C).obj X = X
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@[simp]
theorem category_theory.functor.id_map {C : Type u₁} [category_theory.category C] {X Y : C} (f : X ⟶ Y) :
(𝟭 C).map f = f
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def category_theory.functor.comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) :
C ⥤ E

F ⋙ G is the composition of a functor F and a functor G (F first, then G).

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Instances for category_theory.functor.comp
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@[simp]
theorem category_theory.functor.comp_obj {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) (X : C) :
(F ⋙ G).obj X = G.obj (F.obj X)
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@[simp]
theorem category_theory.functor.comp_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) {X Y : C} (f : X ⟶ Y) :
(F ⋙ G).map f = G.map (F.map f)
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@[protected]
theorem category_theory.functor.comp_id {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) :
F ⋙ 𝟭 D = F
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@[protected]
theorem category_theory.functor.id_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) :
𝟭 C ⋙ F = F
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@[simp]
theorem category_theory.functor.map_dite {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) {X Y : C} {P : Prop} [decidable P] (f : P → (X ⟶ Y)) (g : ¬P → (X ⟶ Y)) :
F.map (dite P (λ (h : P), f h) (λ (h : ¬P), g h)) = dite P (λ (h : P), F.map (f h)) (λ (h : ¬P), F.map (g h))
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@[simp]
theorem category_theory.functor.to_prefunctor_obj {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) (X : C) :
F.to_prefunctor.obj X = F.obj X
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@[simp]
theorem category_theory.functor.to_prefunctor_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) :
F.to_prefunctor.map f = F.map f
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@[simp]
theorem category_theory.functor.to_prefunctor_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) :
F.to_prefunctor ⋙q G.to_prefunctor = (F ⋙ G).to_prefunctor